arXiv:0807.4527v1 [hep-th] 28 Jul 2008 SPhT-T08/118 Imperial/TP/08/DW/02 T-duality, Generalized Geometry and Non-Geometric Backgrounds Mariana Gra˜ na a , Ruben Minasian a , Michela Petrini b and Daniel Waldram c a Institut de Physique Th´ eorique, CEA/Saclay 91191 Gif-sur-Yvette Cedex, France b LPTHE, Universit´ es Paris VI et VII, Jussieu 75252 Paris, France c Department of Physics and Institute for Mathematical Sciences Imperial College London, London, SW7 2BZ, U.K. Abstract We discuss the action of O(d, d), and in particular T-duality, in the context of gener- alized geometry, focusing on the description of so-called non-geometric backgrounds. We derive local expressions for the pure spinors descibing the generalized geometry dual to an SU (3) structure background, and show that the equations for N =1 vacua are invariant under T-duality. We also propose a local generalized geometri- cal definition of the charges f , H , Q and R appearing in effective four-dimensional theories, using the Courant bracket. We then address certain global aspects, in particular whether the local non-geometric charges can be gauged away in, for in- stance, backgrounds admitting a torus action, as well as the structure of generalized parallelizable backgrounds. May 28, 2018
Transcript
arX
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4527
v1 [
hep-
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Jul 2
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SPhT-T08/118
Imperial/TP/08/DW/02
T-duality, Generalized Geometry and Non-Geometric
Backgrounds
Mariana Granaa, Ruben Minasiana, Michela Petrinib and Daniel Waldramc
aInstitut de Physique Theorique, CEA/Saclay91191 Gif-sur-Yvette Cedex, France
b LPTHE, Universites Paris VI et VII, Jussieu75252 Paris, France
cDepartment of Physics and Institute for Mathematical SciencesImperial College London, London, SW7 2BZ, U.K.
Abstract
We discuss the action of O(d, d), and in particular T-duality, in the context of gener-alized geometry, focusing on the description of so-called non-geometric backgrounds.We derive local expressions for the pure spinors descibing the generalized geometrydual to an SU (3) structure background, and show that the equations for N = 1vacua are invariant under T-duality. We also propose a local generalized geometri-cal definition of the charges f , H, Q and R appearing in effective four-dimensionaltheories, using the Courant bracket. We then address certain global aspects, inparticular whether the local non-geometric charges can be gauged away in, for in-stance, backgrounds admitting a torus action, as well as the structure of generalizedparallelizable backgrounds.
Recent developments in flux compactifications brought T-duality to the center stage [1–10].Given a background with isometries, T-duality is a very effective tool for generating new back-grounds. Due to the mixing of the metric and NS two-form B, it can relate string backgroundswith drastically different properties.
A well known example is the action of a single T-duality along an isometry direction of amanifold with an H flux. At the local level, T-duality exchanges the off-diagonal components ofthe metric with those of the B field. Since the Killing vector generating the isometry must beglobally defined, the manifold can be thought of as a circle fibration over a baseM . Topologicallythe fibration can be characterized by the first Chern number. Another topological number isgiven by the integral over a two-cycle on the base of the two-form obtained by contractingthe H-flux with the Killing vector. T-duality exchanges these numbers, leading to a change intopology [11]. Indeed, the dual manifold is again a circle fibration over the same base, but hasin general a different first Chern number.
For backgrounds with d isometries, since the T-duality group O(d, d) is much larger thanthe group of diffeomorphisms, it is not surprising that the action of T-duality can be moreinteresting and complicated [12].
At the level of string sigma model there is a well-defined procedure of performing the dualitytransformations (see e.g. [13]). Given a background with isometries, one gauges them and addsLagrange multipliers. Integrating out the original directions of isometries, while leading to thesame two-dimensional quantum field theory, yields a different target space. There are howeverglobal obstructions in performing the above procedure [14,15]. In order to perform T-duality (inthree or more directions) the component of the H flux fully lying in the directions to be dualizedhas to vanish, in other words, the B-field must respect the isometries. Also, the component of Hwith two legs along the duality directions must be trivial in cohomology, i.e. the corresponding
1
component of the B-field is globally defined. A priori, when such global obstructions are present,T-duality is not possible.
However, it has been proposed in [14], that in some of the obstructed cases, T-duality canlead to consistent string backgrounds. While it is still possible to give local expressions for themetric and B-field, globally the resulting background will not have a conventional descriptionas a good internal manifold, thus the terming “non-geometric compactification”.
The interest in such “non-geometric” backgrounds is also motived by the analysis of four-dimensional effective theories. From the point of view of gauged supergravities in four dimen-sions, potentials governed by a large duality group seem to admit various minima, many ofwhich cannot correspond to conventional string compactifications. It is interesting to identifythose that can be lifted to full string solutions, and the backgrounds obtained via obstructedT-duality transformations are natural candidates. There is much recent work supporting thispossibility [5, 6]. When trying to lift solutions of four-dimensional gauged supergravities toten dimensions, the origin of the gauged symmetries as well as that of the structure constantsin their algebra need to be given a string theory interpretation. For compactifications on d-dimensional homogeneous parallelizable manifolds (loosely called twisted tori), there are 2dsymmetries corresponding to translations and to gauge transformations on B. The twisting ofthe frame bundle and the H-flux appear as structure constants of the “Kaloper-Myers alge-bra” [16]. This algebra is however not covariant under the duality group O(d, d), since suchan invariance would require twice as many charges. It has been argued that the missing halfcorresponds to “non-geometric fluxes”, encoding, for example, monodromies in the T-dualitygroup, mixing metric and B-field [2, 3].
The T-duality group O(d, d) is also the structure group of the generalized tangent bundle,which combines the tangent and cotangent bundle of a d-dimensional manifold in GeneralizedGeometry [17, 18]. The generalized metric on the generalized bundle encodes the informationabout the metric and B-field of the manifold, which are exchanged by T-duality. Additionally,non-trivial patchings of the B-field are naturally incorporated in generalized geometry by defin-ing an extension of the tangent bundle by the cotangent one. In this extension, the patchingbetween two overlapping regions uses, besides the usual diffeomorphisms, an abelian subgroupof O(d, d) involving the B-field. This suggests the use of generalized geometry to describe theaction of T-duality [18,19].
Generalized (complex) geometry is very well suited to describe N = 1 supersymmetric com-pactification with non trivial fluxes. When there are nowhere vanishing spinors on the manifold,one can construct bispinors by tensoring two O(d) spinors with the same and with opposite chi-ralities. These O(d, d) spinors also carry the information about the metric and B-field on themanifold. Each O(d, d) spinor corresponds to an algebraic structure, namely, for six-dimensionalmanifolds, an SU(3, 3) ⊂ O(6, 6). The pair of SU(3, 3) structures defines an SU (3) × SU (3)structure on the generalized tangent bundle. In the context of flux compactifications this cancan be understood as two independent SU(3) structures, one on the left and one on the rightmoving sector. The corresponding globally defined O(d) spinors are the internal supersymme-try parameters. The conditions for N = 1 vacua reduce to a couple of first order differentialequations for the pure spinors, implying the closure of one pure spinor and relating the failureof integrability of the other to the RR fluxes [20]. A manifold admitting a closed pure spinor isa Generalized Calabi-Yau.
Even though generalized geometry conventionally describes ordinary geometrical back–grounds, we will argue that it is still a suitable language to describe some aspects of non-geometric backgrounds. Specifically one is interested in the cases where the compactificationremains locally a manifold, such as arises from obstructed T-duality of conventional geometries(some progress in this direction has been done in [8, 21]).
In this paper we discuss the action of O(d, d) as well as the emergence of the extendedKaloper-Myers algebra, in the context of generalized geometry. We will first do this at the local
2
level, and then discuss the global properties. It is at this point that the difference betweengeometry and non-geometry appears. At the local level, we find the T-duality action on theobjects defining the generalized metric, namely the generalized vielbeins in the generic case, andthe pure spinors in the case of reduced structure. We will mostly consider situations where theO(d, d) transformations are along isometries of the background. We will assume that the B-fieldrespects the isometries and yet is not globally well-defined (in particular the components to befully T-dualized). In these cases we argue that T-duality yields perfectly good local expressions(mixing the metric and the B-field).
We also show that the charges in the extended Kaloper-Myers algebra have a simple inter-pretation as elements of a generalized spin-connection, or, equivalently, as structure constants(or rather functions, in the generic case) of the Courant algebra of generalized vielbeins. Assuch, the distention between geometric and non-geometric charges depends on the frame, andtherefore loses physical content. However, when turning to global properties, we show that thetransformation taking from one frame to another can be ill-defined if there are non-contractibleloops. In this case the non-geometric charges cannot be globally gauged away. The distinc-tion between geometric and non-geometric situations is nicely rephrased in terms of right andleft mover sectors of string theory. Generalized geometry suggests the use of a different set ofvielbeins for the left and right movers, which transform nicely under O(d, d).1 For geometricbackgrounds it is always possible, after T-duality, to perform a well defined local O(d) × O(d)transformation to set the vielbeins for the right and left moving sectors to be the same every-where. On the contrary, when such a transformation is only possible locally, the backgroundis non-geometric. A similar situation arises in the doubled formalism, where for geometricalbackgrounds one is again able to use O(d)×O(d) transformations to write the doubled vielbeinsin a particular triangular form [10].
The two different sets of vielbeins on right and left movers have nice transformation proper-ties under O(d, d) . Moreover they provide a way to determine the O(d, d) transformation of anSU (3)× SU (3) structure. In principle, given the two new SU (3) structures, one should be ableto build the corresponding T-dual spinors. However, since the pure spinors are mixtures of leftand right moving sectors, determining them explicitly can be quite challenging. Here we use adifferent approach and we study directly the action of O(d, d) on the spinors. In particular wederive local expressions for the pure spinors dual, or mirror, to those corresponding to a singleSU (3) structure (i.e. where the original structures on the left and on the right sector coincide).These correspond generically to SU (3) × SU (3) structures.
As already mentioned, generalized geometry allows for an elegant classification of Type IIflux background with N = 1 supersymmetry. Given that supersymmetry equations are local,they could still be considered in the case of non-geometric backgrounds, when these admit a localdescription. In other words, one might wonder whether considering T-duality along isometriesthat commute with the supersymmetry generators, one might still have good local solutions,also when T-duality is obstructed. We show that the N = 1 supersymmetry equations on purespinors are invariant under T-duality.
A very simple example of a generalized geometrical background is one where there is aglobally defined set of generalized vierbein, the analogue of a conventional parallelizable back-ground. The Courant bracket on the preferred frame then provides a natural global definition ofthe generalized charges. Many of the simplest non-geometrical examples are of this “generalizedparallelizable” type. We discuss some necessary conditions on the local geometry in this caseand in particular show that the R charge always vanishes.
The paper is organized as follows. In Section 2 we review the necessary ingredients ofgeneralized geometry, and find the O(d, d) transformations of the vielbeins. In Section 3 wediscuss how T-duality acts on the generalized structures, and find explicit expressions for the
1Two sets of vielbeins on the target space were introduced in the context of T-duality in [22]. They alsoappear naturally in the doubled formalism [1].
3
duals of an SU(3) structure. We also show that the equations for N = 1 vacua are invariantunder T-duality. In Section 4 we introduce the generalized spin connection and we discusshow the charges of the extended Kaloper-Myers algebra arise locally from the Courant bracket.Finally in Section 5 we discuss global issues and non-geometricity, as well as the structure ofgeneralized parallelizable backgrounds.
2 Generalized geometry
This section starts with a review of generalized geometry, the generalized O(d, d) spinors andthe generalized metric H, which encodes the ordinary metric g and the B-field. The generalizedmetric defines an O(d) × O(d) structure, and we also introduce a natural set of generalizedvielbeins for H. This latter has been previously analyzed by Hassan [22]. One new element hereis the discussion of how the dilaton naturally enters the definition of O(d, d) spinors.
We then turn, in the context of six-dimensional manifolds, to the various definitions ofSU (3)× SU (3) structures relevant to supersymmetric backgrounds.
Of particular interest is how the O(d, d) group acts on the generalized vielbeins and henceon the ordinary vielbein and B-field (also discussed in [22]). In addition we consider the actionof O(6, 6) on an SU (3) × SU (3) structure. These will be useful in the following parts of thepaper were we specialize to T-duality transformations in a number of specific cases. Here weconsider only the action of O(d, d) at a point in the manifold.
2.1 Generalized tangent bundle
The basic idea of generalized geometry [17,18] is to combine vectors and one-forms into a singleobject. Formally, on a d-dimensional manifold M one introduces the generalized tangent bundleE which is a particular extension of T by T ∗
0 −→ T ∗M −→ Eπ−→ TM −→ 0. (2.1)
Sections of E are called generalized vectors. Locally they can be written as X = x + ξ wherex ∈ TM and ξ ∈ T ∗M . In going from one coordinate patch Uα to another Uβ, we have to firstmake the usual patching of vectors and one-forms, and then give a further patching describinghow T ∗M is fibered over TM in E. This gives
x(α) + ξ(α) = a(αβ)x(β) +[
a−T(αβ)ξ(β) − ia(αβ)x(β)
ω(αβ)
]
, (2.2)
where a(αβ) ∈ GL(d,R), ω(αβ) is a two-form and a−T = (a−1)T . Using a two-component notationto distinguish the vector and form parts of X we can write
X(α) =
(
x(α)ξ(α)
)
=
(
1 0ω(αβ) 1
)
(
a(αβ) 0
0 a−T(αβ)
)
(
x(β)ξ(β)
)
= p(αβ)X(β) . (2.3)
In fact one makes the further restriction that ω(αβ) = −dΛ(αβ), where Λ(αβ) are required tosatisfy
Λ(αβ) + Λ(βγ) + Λ(γα) = g(αβγ)dg(αβγ) (2.4)
on Uα ∩ Uβ ∩ Uγ and gαβγ := eiα is a U(1) element. This is analogous to the patching of aU(1) bundle, except that the transition “functions” are one-forms, Λ(αβ). Formally it is calledthe “connective structure” of a gerbe. The point is that it is the geometrical structure oneneeds to introduce B, the two-form analogue of an ordinary one-form U(1) connection, with acorrespondingly quantized field strength H.
Given the split into vectors and forms, there is a natural O(d, d)-invariant metric η on E,given, on each patch, by
η(X,X) = ixξ, (2.5)
4
or, in two-component notation, η(X,X) = XT ηX with
η =1
2
(
0 1
1 0
)
. (2.6)
The metric is invariant under O(d, d) transformations acting on the fibres of E. A generalelement O ∈ O(d, d) can be written in terms of d× d matrices a, b, c, and d as
O =
(
a bc d
)
, (2.7)
under which a generic element X ∈ E transforms by
X =
(
xξ
)
7→ OX =
(
a bc d
)(
xξ
)
. (2.8)
The requirement that η(OX,OX) = η(X,X) implies aT c + cTa = 0, bTd + dT b = 0 andaTd + cT b = 1. Note that the GL(d) action on the fibres of TM and T ∗M embeds as asubgroup of O(d, d). Concretely it maps
X 7→ X ′ =
(
a 00 a−T
)(
xξ
)
, (2.9)
where a ∈ GL(d). Similarly the first factor in (2.3) is also an (Abelian) subgroup GB ⊂ O(d, d).Given a two-form ω, we write
eω =
(
1 0ω 1
)
such that X = x+ ξ 7→ X ′ = x+ (ξ − ixω). (2.10)
This is usually referred to as a B-transform. Given a bivector β one can similarly define anotherAbelian subgroup of β-transforms
eβ =
(
1 β0 1
)
such that X = x+ ξ 7→ X ′ = (x+ β · ξ) + ξ. (2.11)
The patching (2.3) of E was by elements of GL(d) and GB . Together these form a subgroupwhich is a semi-direct product Ggeom = GB⋊GL(d). A general element of Ggeom can be writtenas
p = eω(
a 00 a−T
)
=
(
a 0ωa a−T
)
. (2.12)
This patching means that the structure group of the generalized tangent space E actuallyreduces from O(d, d) to Ggeom. The embedding of Ggeom ⊂ O(d, d) is fixed by the projectionπ : E → TM . It is the subgroup which leaves the image of the related embedding T ∗M → Einvariant.
There is also a natural bracket on generalized vectors known as the Courant bracket, whichencodes the differentiable structure of E and will play an important role in what follows. It isdefined as
where [x, y]Lie is the usual Lie bracket between vectors and Lx is the Lie derivative. TheCourant bracket is invariant under the action of elements of Ggeom, (2.12), where the GL(d)transformations a are generated by diffeomorphisms and the B-shifts ω are closed, dω = 0.
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2.2 Generalized metrics, generalized vielbeins and O(d)×O(d) structures
In the generalized geometry picture the metric g and the B-field combine into a single objectwhich defines an O(d)×O(d) structure on E. To define an O(d)×O(d) structure we need thebundle E to split into two orthogonal d-dimensional sub-bundles E = C+ ⊕ C− such that themetric η decomposes into a positive-definite metric on C+ and a negative-definite metric on C−.The subgroup of O(d, d) which preserves each metric separately is then O(d)×O(d). Since anyelement of E which is a pure vector or a pure one-form is null with respect to η, such elementscannot lie in C+ or C−. Hence we can write a generic element X+ ∈ C+ as x + Mx, wherex ∈ TM and, in components, the form part is given by Mmnx
n for some general matrix M .(This actually describes an isomorphism between TM and C+.) If we write Mmn = Bmn+gmn,where g is symmetric and B antisymmetric, we see that the patching condition (2.3) impliesthat
g(α) = g(β), B(α) = B(β) − dΛ(αβ), (2.14)
and hence is associated to the connective structure of a two-form B-field. Orthogonality betweenC+ and C− implies that a generic element of X− ∈ C− can be written as X− = x+ (B − g)x.
Another way to define this structure is to introduce the O(2d)-invariant generalized metric 2
H = η|C+− η|C−
. (2.15)
Writing a general element X = x + ξ ∈ E as X = X+ +X− with X± = x± + (B ± g)x± onefinds that the generalized metric H takes the form
H =
(
g −Bg−1B Bg−1
−g−1B g−1
)
. (2.16)
We can also introduce generalized vielbeins, where the local Lorentz symmetry is replacedby O(d)×O(d). They parametrise the coset O(d, d)/O(d)×O(d) and encode the metric g andthe B-field. There are many different conventions one could use. Consider a basis of generalizedone-forms EA ∈ E∗ with A = 1, . . . 2d. (Note that η gives an isomorphism between E and E∗ sowe can equally well think of the EA as generalized vectors.) One possibility is then to requirethat the metrics η and H take the form
η = ET
(
1 00 −1
)
E, H = ET
(
1 00 1
)
E. (2.17)
Explicitly we have
E =1√2
(
e+ − eT+B eT+−e− − eT−B eT−
)
=1√2
(
eT+(g −B) eT+−eT−(g +B) eT−
)
, (2.18)
where we have introduced two sets of (ordinary) vielbeins ea± and their inverse e± a satisfying
g = eT±e± or gmn = ea±meb±nδab ,
g−1 = e±eT± or gmn = em± ae
n± bδ
ab,(2.19)
and e±e± = e±e± = 1. With these conventions, the first d generalized vielbeins form a basisfor C+ and the second d form a basis for C−. The local O(d)×O(d) action simply rotates eachset of vielbeins. Concretely we can write
E 7→ KE , K =
(
O+ 00 O−
)
with O± ∈ O(d) . (2.20)
2 In [18], the O(d, d) invariant generalized metric is defined via the product structure G = −J1J2, given twocommuting generalized almost complex structures. This is related to our definition by H = ηG.
6
In type II string theory compactified on a six-dimensional manifold M , the subbundles C±
have a natural interpretation in terms of the world-sheet theory: they are associated to theleft and right mover sectors; e± are the corresponding vielbeins. The spinors transform underone or the other of the O(d) groups. It is then usual to choose e+ = e− so that the samespin-connections appear, for instance, in the derivatives of the two gravitini. However, this is,of course, not strictly necessary.
From the O(d, d) action on the generalized metric and vielbein it is straightforward to recoverthe familiar O(d, d) transformations on the metric, B-field and vielbein. The generalized metric(2.16) transforms under O(d, d) as
H → H′ = OTHO , (2.21)
with H and O given in (2.16) and (2.7), respectively. Given this transformation, we can derivethe transformation of the bases e± under O(d, d). The generalized basis forms EA transform asE 7→ EO, and hence the vielbeins transform as
e+ 7→[
dT + bT (B + g)]
e+ ≡ ˆe+
e− 7→[
dT + bT (B − g)]
e− ≡ ˆe− .(2.22)
This agrees with the result given in [22]. Note that, if we initially set e+ = e−, generically thisis no longer true after the O(d, d) transformation, and one must make a compensating Lorentztransformation to restore the condition.
It is possible to use a different set of conventions where η and H take the form
η = ET
(
0 1
1 0
)
E , H = ET
(
1 00 1
)
E . (2.23)
In this basis the generalized vielbein can be written as
As before, one can always make an O(d)×O(d) transformation to set e+ = e− = e and put thegeneralized vielbein into the triangular form
E =
(
e 0−eTB eT
)
. (2.26)
Note that in these conventions the vielbeins (2.24) are not a natural basis for C± since they donot diagonalise the O(d, d) metric η. However they will be of particular interest in this paperbecause the latter form (2.26) is invariant under the Ggeom subgroup of O(d, d) transformations.
2.3 O(d, d) spinors
Given the metric η, one can define Spin(d, d) spinors. These are Majorana–Weyl and we writethe two helicity spin bundles as S±(E). Locally, the Clifford action of X ∈ E on the spinorscan be realized as an action on forms Φ ∈ Λeven/oddT ∗M
∣
∣
Uαgiven by
X · Φ := (xmΓm + ξmΓm)Φ = ixΦ+ ξ ∧ Φ , (2.27)
7
where Γ, Γ are the O(d, d) gamma matrices. It is easy to see that
(XY + Y X) · Φ = 2η(X,Y )Φ , (2.28)
as required. One also finds that, in going from one patch to another, the patching of E impliesthat
Φ±(α) = edΛ(αβ)Φ±
(β) , (2.29)
where the exponentiated action is by wedge product. Note that the usual action of the exteriorderivative on the component forms is compatible with this patching and gives an action
d : S±(E) → S∓(E) . (2.30)
In terms of the Spin(d, d) group one can view this as a Dirac operator taking positive helicityspinors to negative helicity spinors and vice versa.
Let us now return to the GL(d) action (2.9) on the tangent and cotangent bundles. If wetake an infinitesimal transformation with a = 1+ θ + . . . , the induced action on the spinors isgiven by
δΦ = 12θ
mn
(
ΓmΓn − ΓnΓm
)
Φ . (2.31)
The degree of the component forms in Φ remains unchanged: in particular, each form transformsas
δΦm1...mp = −p θn[m1Φ|n|m2...mp] +
12θ
nnΦm1...mp . (2.32)
The first term correctly describes the transformation of an element of ΛpT ∗M under GL(d).The second term however corresponds to a rescaling of the form by a factor of |det a|1/2. Thisimplies that we should locally identify [18]
Φ ∈ |ΛdT ∗M |−1/2 ⊗ Λeven/oddT ∗M∣
∣
∣
Uα
. (2.33)
However, this presents a predicament: we cannot define the exterior derivative on such ob-jects, because the extra |ΛdT ∗M |−1/2 factor breaks the diffeomorphism symmetry. One solutionis to identify
Φ ∈ L⊗ Λeven/oddT ∗M∣
∣
∣
Uα
, (2.34)
where we have introduced a new (trivial) real line bundle L with sections e−φ ∈ L that transformas
e−φ 7→ |det a|1/2e−φ (2.35)
under the GL(d) action on TM , but which transform as scalars under diffeomorphisms. Wehave suggestively written the sections of L as e−φ since we will see in the next section that theten-dimensional dilaton indeed transforms in this way.
Under the other two elements of O(d, d) discussed in the previous section, Eqs. (2.10) and(2.11), the spinor representation transforms
Φ± 7→ eω+βΦ± , (2.36)
where ω acts by wedge product and β by contractions.
Using the generalized vielbeins (2.18) one can also introduce a basis for the O(d, d) gammamatrices Γ, Γ adapted to the O(d)×O(d) structure. One defines
(
Γ+
Γ−
)
= (E−1)T(
Γ
Γ
)
=
eT+
(
Γ + (g −B)Γ)
eT−
(
Γ− (g +B)Γ)
, (2.37)
8
which satisfy
Γ+a ,Γ
−b = 0, Γ+
a ,Γ+b = 2δab, Γ−
a ,Γ−b = −2δab . (2.38)
One can then decompose the Spin(d, d) spinors into Spin(d) × Spin(d) objects. If d is even wecan write
Γ+a = γa ⊗ 1, Γ−
a = iγ ⊗ γa , (2.39)
where γa are Spin(d) gamma matrices and γ = γ(d) = γ1 . . . γd if d/2 is even and γ = −iγ(d) ifd/2 is odd, so that γ2 = 1. Similar expressions can be written when d is odd. The correspondingdecompositions of the Spin(d, d) spinors are
where η1± and η2± are chiral Spin(d) spinors satisfying γ η± = η±.
The generalized metric allows us to relate the O(d) × O(d) decomposition of the O(d, d)spinors to the GL(d) decomposition (2.34). It is easiest to start by choosing the vielbeins suchthat e+ = e−. This identifies a common O(d) subgroup of O(d) × O(d): η1+ and η2+ are nowspinors of the same group so that, under this group, Φ± is a spinor bilinear. However, any spinorbilinear can be expanded as a sum of forms using products of gamma matrices. In particular
η1+η2+ =
1
nd
∑
p even
1
p!
(
η2+γm1...mpη1+
)
γmp...m1 ,
η1+η2− =
1
nd
∑
p odd
1
p!
(
η2−γm1...mpη1+
)
γmp...m1 ,(2.41)
where γm are nd×nd matrices, and we have used the metric gmn to write the component formsin tangent space indices. Given an expansion of the form (2.41), the Clifford action on Φ± is
X · Φ± = 12 [x
mγm,Φ±]∓ + 12 [ξmγm,Φ±]± , (2.42)
where Φ± are defined in (2.44).
Note, however, that the forms (2.41) are neither twisted with dΛαβ, as in (2.29), nor trans-form with the additional factor of |det a|1/2 under GL(d). If we use the short-hand that η1+η
2±
represent the corresponding sums of forms as in (2.41), naively we find that the decompositionof Φ± under GL(d) is related to the bispinor by
However, this identifies O(d, d) spinors as sections of (2.33), which precludes the use of theexterior derivative. Introducing the line bundle L we can take Φ± to be sections of (2.34), andinstead have
Φ+ = e−φe−Bη1+η2+, Φ− = e−φe−Bη1+η
2− , (2.44)
where e−φ is some section of L. By construction
e2φ/√
det g (2.45)
is invariant under O(d, d). This is precisely the way the ten-dimensional dilaton transforms.Thus we see that the dilaton appears very naturally in generalized geometry: together with thegeneralized metric H, encoding g and B, the dilaton defines the isomorphism between S±(E)and Λeven/oddT ∗M .
Finally we note that an O(d, d) spinor is said to be pure if it is annihilated by half of thegamma matrices (or equivalently if its annihilator is a maximally isotropic subspace of E). Anypure spinor can be represented as a wedge product of an exponentiated complex two-form with
9
a complex k-form. The degree k is called type of the pure spinor, and, when the latter is closed,it serves as a convenient way of characterizing the geometry.
A pure spinor defines an SU (d, d) structure on E. A further reduction of the structure groupto SU(d) × SU(d) is given by the existence of a pair of compatible pure spinors. Two purespinors are said to be compatible when they have d/2 common annihilators. By construction,the spinors (2.44) are pure and also compatible.
3 T-duality and SU (3)× SU (3)-structures
In this section we would like to address the question of how T-duality acts on backgrounds withSU (3) × SU (3) structure. Such geometries describe string compactifications leading to N = 2effective theories in four dimensions and can be defined by a pair of O(6, 6) spinors. We shallgive explicit expressions for the new SU (3) × SU (3) structure in two natural cases. First wegive the transformation of the structure when the original manifold is a T 3 fibration and weperform three T-dualities, that is the map to the mirror configuration. Then we consider thesimpler case where the original manifold is a T 2 fibration and we perform a pair of T-dualities.In each case, we start with a given SU (3) structure with non-trivial H-flux. We shall see inparticular that T-duality can change the type of structure.
Furthermore, in some cases we will find that naively the structure is ill-defined. We discussthis feature in detail for the T 2-fibrations, and argue that it arises precisely when the dualbackground is non-geometrical. The analysis is entirely consistent with the original discussionsof non-geometry for such fibrations [3, 14]. Here we focus our attention on the transformationof the additional SU (3)× SU (3) structure.
Finally, we will also show that T-duality maps supersymmetric SU (3)×SU (3) backgroundsto supersymmetric SU (3)×SU (3) backgrounds. This requires that the Lie derivative along theT-duality direction of the pair of O(6, 6) spinors defining the geometry vanishes.
The section begins with a general discussion of T-duality in the context of generalizedgeometry. This leads to a simple expression for the action of T-duality on O(d, d) spinors whichare the defining objects for SU (3) × SU (3) structures. We then review the relation betweenSU (3)×SU (3) structures and supergravity backgrounds, before turning to considering T-dualityon them. We conclude with the analysis of T-duality on supersymmetric backgrounds.
3.1 Generalized Lie derivative, generalized Killing vectors and T duality
In string theory T-duality is a non-local transformation. However, at the level of supergravity,there is a corresponding transformation, given by the Buscher rules [23], which can be viewedas a local transformation of the supergravity fields, taking solutions to solutions. In this sectionwe discuss how this local T-duality acts on the generalized structure. We will see that formallyit is simply a O(d, d) gauge transformation on E.
Buscher rules apply when one has a supergravity background that admits a Killing vectorfield v satisfying3
Lvg = LvH = 0 . (3.1)
The condition LvH = 0 implies that locally one can make a gauge transformation, B′ = B+dζ ′,such that LvB
′ = 0 or, equivalently, LvB − dζ = 0, where ζ = −ivdζ′ + df . Buscher rules are
then applied to the gauge transformed background (g,B′) and generate a new background(g, B). Thus, in a generic gauge we require
Lvg = 0 ,
LvB − dζ = 0 ,(3.2)
3If there is also a non-zero Ramond–Ramond flux, one must further require that the Lie derivative of the fluxvanishes.
10
so that to define the T-duality action on the supergravity fields we really need a pair (v, ζ).From the action (3.2) on B we see that (v, ζ) act as an infinitesimal diffeomorphism generated
by v together with a gauge transformation. Writing V = v+ ζ, we can define the correspondingaction on sections X = x+ ξ of E as a sort of “generalized Lie derivative”
LV X = [v, x]Lie + (Lvξ − ixdζ) , (3.3)
where [v, x]Lie is the Lie bracket and Lv is the ordinary Lie derivative. This combination of Vand X is actually none other than the Dorfman bracket [18,24] whose antisymmetrization givesthe Courant bracket (2.13). It is the derived bracket for the exterior derivative d.
Note that this action is very natural given the bundle structure (2.1). We naturally identifyas equivalent bundles E which are related by diffeomorphisms of the manifold M and gaugetransformations which preserve the patching (2.3). Infinitesimally, together these are equivalentto an action of the generalized Lie derivative.
Given this definition of LV on generalized vectors, it is then natural to define the generalizedLie derivative of H by
This is in analogy to the construction for a conventional Lie derivative and here, when actingon a scalar function such as H(X,Y ), we define LV f = Lvf = ivdf . It is then easy to see that
LV H =
Lvg − (LvB − dζ)g−1B−B(Lvg
−1)B −Bg−1(LvB − dζ)(LvB − dζ)g−1 +B(Lvg
−1)
−g−1(LvB − dζ)− (Lvg−1)B Lvg
−1
. (3.5)
(Note that a similar calculation implies for the O(d, d) metric (2.5) that LV η = 0.) Therequirement (3.2) on (g,B) then simply translates into
LV G = 0 , (3.6)
or, in other words, that V defines a “generalized Killing vector”.Given a generalized Killing vector V , we can then define the corresponding Buscher duality
as follows. First recall that there was really an ambiguity in V = v + ζ, since the generalizedLie derivative only depends on dζ so we can always shift ζ by df for an arbitrary function f .Using this freedom we can always normalize V
η(V, V ) = 1 . (3.7)
Concretely, for any vector field v we can introduce a coordinate t such that v = ∂/∂t. Inaddition, from (3.1), we know we can write ζ = −ivdζ
′ + df . Setting f = t we have
V = ∂/∂t + (dt− i∂/∂tdζ′) , (3.8)
and hence η(V, V ) = 1. We then construct the O(d, d) element
TV = 1− 2V V T η . (3.9)
The condition η(V, V ) = 1 implies that η(TV X,TV X) = η(X,X) so TV ∈ O(d, d) and, inaddition, T 2
V = 1. We can choose local bases on TM and T ∗M such that, if e1 = v = ∂/∂t isthe first basis element of TM and its dual one-form e1 = dt is the first element for T ∗M . Thentaking ζ ′ = 0, the T-duality matrix reads
Te1+e1 =
(
1−m mm 1−m
)
, m =
1 0 ... 00 0 ... 0...
.... . .
...0 0 ... 0
. (3.10)
11
The T-dual generalized metric H is simply given by
H(X,X) = H(TV X,TV X) , (3.11)
or H = T TV HTV . The action on O(d, d) spinors is by an element of Pin(d, d) equal simply to
the Clifford action of VΦ = TV Φ = i∂/∂tΦ+ ζ ∧ Φ , (3.12)
where ζ = dt− i∂/∂tζ′.
Note that T-duality is usually defined in the gauge where the NS two-form is given byB′ = B + dζ ′ and hence satisfies LvB
′ = 0. Here we see that TV can be written as
TV = edζ′ · TV0 · e−dζ′ , (3.13)
where V0 = ∂/∂t + dt. Thus the action of TV is to first make a gauge transformation on H toset the NS two form to B′, and then act by conventional T-duality. Note also that, as alwaysfor Buscher duality, the choice of coordinate t used to write v = ∂/∂t is not unique. However,the effect after T-duality is simply an additional gauge transformation.
3.2 SU (3)× SU (3) structures and supergravity
In type II supergravities compactified on a six-manifold M , the two supersymmetry parametersdecompose into two chiral Spin(6) spinors transforming under the Spin(6) groups associatedwith C+ and C−, respectively. When considering either a supersymmetric background, ora background leading to a low-energy supersymmetric effective action (such as a Calabi–Yaumanifold with non-zero fluxes), the supersymmetry picks out a particular pair of globally defined,nowhere vanishing spinors (η1+, η
2+).
Since Spin(6) ≃ SU (4), a single spinor η+ is invariant under an SU (3) subgroup of Spin(6).If η+ is globally defined and nowhere vanishing, it defines an SU (3) structure. This is a topo-logical restriction: the tangent bundle TM is patched using only SU (3) transformations. It isequivalent to the existence of a pair of globally defined, nowhere vanishing forms J ∈ Λ2T ∗Mand Ω ∈ Λ3T ∗
C
M . Thus the pair of spinors (η1+, η2+) defines a pair of SU (3) structures. More
precisely they are invariant under an SU (3)×SU (3) subgroup of O(6, 6), and we say they definean SU (3) × SU (3) structure. Note that the common subgroup preserving both η1+ and η2+ isgenerically SU (2), though at points where they are parallel it becomes SU (3); in this sense η1+and η2+ define a “local” SU (2) structure.
Thus we see that the SU (3) × SU (3) structure can be defined in a number of equivalentways:
(a) the generalized metric H (defining g and B) together with the pair (η1+, η2+);
(b) the two pairs of SU (3) structures (J+,Ω+) and (J−,Ω−) together with B;
(c) the (local) SU (2) structure, together with a complex scalar η1+η2+ and B;
(d) a pair of complex generalized spinors Φ± ∈ S±C
(E).
The relations between these various descriptions are as follows. First we fix the normalizationof the spinors: η1+η
1+ = η2+η
2+ = 1. The two SU(3) structures are defined as
J+mn = −iη1+γmnη
1+, J−
mn = −iη2+γmnη2+ , (3.14)
Ω+mnp = −iη1−γmnpη
1+, Ω−
mnp = −iη2−γmnpη2+ . (3.15)
Here and in all the following definitions γm are Spin(6) gamma matrices and γ(7) = −iγ1 . . . γ6.These two SU(3) structures are defined on C+ and C−, respectively. As such they can be alwayswritten in a standard form in terms of the vielbeins e±
Locally the two SU (3) structures define an SU (2) structure. The latter is defined by acomplex one form z = v + iv′, and a triplet of real two-forms (J1, J2, J3), or, equivalently, areal two-form j and a complex two-form ω [25]. One can then always express the two SU (3)-structures in terms of the SU (2) objects, though the decomposition is not unique, since itdepends on the different choices of j within the triplet (J1, J2, J3). Here we will use a decom-position where j is naturally associated to (J+,Ω+) and η2+ = k‖η
1+ + k⊥(v + iv′)mγmη1−. This
gives
J+ = j − i2 z ∧ z , J− = (|k‖|2 − |k⊥|2)J+ +Re(k‖k⊥ω)− 4i|k⊥|2 z ∧ z ,
Ω+ = z ∧ ω , Ω− = k2‖Ω+ − k2⊥ω ∧ z − 4k⊥k‖j ∧ z .
(3.17)
To define the pure spinors we must decompose under the two Spin(6) subgroups of Spin(6, 6).We can realize the O(6, 6) gamma matrices as
Γ+m = γm ⊗ 1 Γ−
m = γ(7) ⊗ γm . (3.18)
Here we are implicitly assuming that e+ = e−. One can use this decomposition to write theO(6) × O(6) spinors as Spin(6) bispinors. For example, if η1+ and η2+ are chiral spinors of thefirst and second Spin(6) group, respectively, we can write
Φ+ = e−φ−Bη1+η2+ ∈ S+(E), Φ− = e−φ−Bη1+η
2− ∈ S−(E). (3.19)
Explicitly the two pure spinors read
Φ+ = e−φ−B+ 12z∧z (k‖e
−ij − ik⊥ω) , (3.20)
Φ− = e−φ−B z (k⊥e−ij + i k‖ω) . (3.21)
3.3 Examples
In this section we first determine the structure of the mirror of a generic manifold with a T 3
fibration, by doing T-dualities along the T 3 fiber. In particular we construct explicitly theresulting mirror local SU (2) structure. Mirror symmetry transformations of pure spinors forSU (3) and SU (2) structure were also studied in [21], by doing Fourier-Mukai transforms of thepure spinors.
As discussed in the previous section, the patching of the B-field, (2.14), induces the patching(2.29) on the spinors so that e−BΦ is globally well defined on E. It is well known that, under asingle T-duality, the components of the B-field with no legs along the T-dualized directions stayunchanged, while those with one leg are exchanged with the connection along the T-dualizedfiber [26]. In that case, spinors are still globally well-defined. Under a second T-duality, however,if the original B-field has a component with both legs along the T-dualized directions, there isno connective structure allowing to define objects globally.
In the rest of this section we focus on the T-duality action for the latter case. We willillustrate this with two simple toroidal examples, where the B-field is purely along the twodirections to be dualized.
All the calculations we perform in this Section are local. We come back to global issues inSection 5.
3.3.1 Mirror symmetry on an T 3-fibered manifold with H-flux
Consider the case of a manifold with a T 3-fibration and generic B field. We assume there isan SU (3) structure such that T-duality on the T 3-fibration corresponds to mirror symmetry,
13
that is that the T 3 fibers are special-Lagrangian. We then act by T-duality and ask what isthe structure of the mirror compactification. For the case of no B-field with two legs along thefiber, this computation was done in [27], where it was found that Φ+ and Φ− are exchangedunder mirror symmetry. Here, with generic B, we find that the new structure is SU (3)×SU (3)rather than SU (3).
We use the same notation as in [27], except that we denote the coordinates of T 3 fibrationby (y1, y2, y3), and those of the base by (x1, x2, x3). The metric and B-field are
ds2 = gijdxidxj + hαβη
αηβ ,
B2 =1
2B
(0)ij dxi ∧ dxj +
1
2B
(1)iα dxi ∧ (dyα + ηα) +
1
2B
(2)αβη
α ∧ ηβ , (3.22)
where ηα ≡ dyα + λαi dx
i and the superindex on B indicates the number of legs along thefiber. The vielbein is (ea
′
i dxi, eaαη
α), where a, a′ = 1, 2, 3 are fiber and base orthonormal indices,respectively
δa′b′ea′
i eb′
j = gij , δabeaαe
bβ = hαβ . (3.23)
The holomorphic vielbeins are
Za = eaα ηα + i δaa′ e
a′
i dxi . (3.24)
The original SU (3) structure is given by the pure spinors 4
Φ+ = e−φ−B−iJ , J =i
2ZaZ a
Φ− = e−φ−BΩ , Ω =1
6ǫabcZ
a ∧ Zb ∧ Zc . (3.25)
The three T-dualities on the fiber are generated by the generalized vectors Vα = ∂/∂yα + dyα.Writing T = TV1TV2TV3 , we have TΦ+ = Φ−, TΦ− = Φ+, where Φ−, Φ+ can be written in theform given in (3.20), (3.21) with
z =1
|B(2)|ǫabcB(2)bc Za , e−φ = e−φ
√
|h+B(2)| ,
=i
2Za ¯Za − i
2z ∧ z , k⊥ = i|B(2)|
√h
√
|h+B(2)|,
ω = − 1
|B(2)|B(2)ab Z
aZb , k‖ =
√h
√
|h+B(2)|, (3.26)
where |B(2)| =√
B(2)ab B
(2)ab , and the dual holomorphic coordinates are
Za = eaαηα + iδaa′ e
a′i dx
i , ηα = dyα + λαi dx
i . (3.27)
The dual (plus) vielbeins for the dual metric and the connection are related to the original onesby
ea′
i = ea′
i , eaα = eaβ
(
(h+B(2))−1)βα
, λαi = B
(1)iα (3.28)
The dual B-field is
B(0) = B(0) , B(1)iα = λα
i , B(2)αβ = −
(
(h+B(2))−1)αλ
Bλρ
(
(h−B(2))−1)ρβ
, (3.29)
as expected from Buscher rules. Note that in orthonormal indices B(2)ab = −B
(2)ab .
4For a generic B-field, B ∧ Ω 6= 0 so strictly speaking the original structure is not SU (3).
14
In the limit of vanishingB(2), we recover the results of [27], namely Φ+ and Φ− get exchangedunder T-duality (if we write them in terms of dual vielbeins), and define a good mirror SU(3)structure. For nonzero B(2), we get a mirror SU (3) × SU (3) structure that can be defined inpatches, but does not appear to make sense globally. In the following we will focus on this issuein more detail in the slightly simpler case of a pair of T-dualities with B-field only on the T 2
fibres.
3.3.2 Two T-dualities on T 2-fibration with H-flux
We now consider the simpler example of a T 2-fibered manifold with SU (3) structure defined by
where ei are a set of vielbeins. We will also assume there is a B-field along the fibre only. Wewill further assume that the fibration is trivial, implying we can introduce coordinates such thatei = ridx
i etc for the fibered directions. It would be straightforward to include a non-trivialfibration but it is well known that this is T-dual to a non-trivial B-field, so we instead considerthe latter.
We will consider two distinct cases, where the T 2-fibration lies along e1 and e4 and e2 ande3, respectively. These two cases are inequivalent with respect to the SU (3) structure.
Not type changing
We consider first the case where the T 2 fibration lies in two directions, e1 and e4, which arepaired by the complex structure. The B-field on the T 2-fiber can be written as
B =b
r1r4e1 ∧ e4 = b dx1 ∧ dx4 . (3.31)
The factor b in the B-field can be a function of the base. For instance if the base was T 4, thatis we compactify by identifying xi ∼ xi + 1, we could take for example b = hx6, correspondingto a flux H146 = h (in coordinate indices).
We now perform two T-dualities along ∂1, ∂4, that is, in the notation of Section 3.1, usingthe two generalized vectors V1 = ∂1 + dx1 and V4 = ∂4 + dx4. We obtain again an SU(3)structure of the form
Φ+ = eiθ+e−φ−B−iJ , J =e1 ∧ e4
b2 + r21r24
+ e2 ∧ e5 + e3 ∧ e6 , (3.32)
Ω =b− ir1r4b2 + r21r
24
(r4r1e1 + i
r1r4
e4) ∧ (e2 + ie5) ∧ (e3 + ie6) , (3.33)
Φ− = eiθ−φ−BΩ , B = − b
r1r4(b2 + r21r24)
e1 ∧ e4 , (3.34)
e−φ = e−φ√
b2 + r21r24 , (3.35)
tanθ± = ∓r1r4b
. (3.36)
where Φ± = TV1TV4(Φ±), are the dual pure spinors. This structure can be rewritten using either
15
the set of ei+ vielbeins or ei−. These read5
e1± =±r24 e1−b
r1r4
e4
|h+B|f= r1
±r24 dx1 − b dx4
|h+B|f
e4± =±r21 e4+b
r4r1
e1
|h+B|f= r4
±r21 dx4 + b dx1
|h+B|f, (3.37)
ei± = ei , i 6= 1, 4 , (3.38)
where |h+B|f = b2 + r21r24 is the determinant of the matrix h+B along the fiber directions e1
and e4. The structure after T-duality is still SU (3), since J+ = J− = J , and Ω+ = Ω− = Ω.
Type changing
We now turn to the next to simplest example. Here we assume the T 2-fibration lies along the e2
and e3 directions, i.e. on two directions not paired by the complex structure. We again assumethat B lies solely along the fibration so that6
B =b
r2r3e2 ∧ e3 = b dx2 ∧ dx3 . (3.39)
Performing two T-dualities generated by V2 = ∂2+dx2 and V3 = ∂3+dx3, we now get a localSU(2) structure on dual space. The structure is defined by the pure spinors Φ+ = TV2TV3(Φ
+),Φ− = TV2TV3(Φ
−), with Φ+, Φ− given in (3.20), (3.21) and where the SU(2) structure can bewritten in terms of ei+
z = −i(e1+ + ie4+) ,
= e2+ ∧ e5+ + e3+ ∧ e6+ , k⊥ = ir2r3
√
b2 + r22r23
,
ω = (e2+ + ie5+) ∧ (e3+ + ie6+) , k‖ =b
√
b2 + r22r23
B = − b
r2r3e2+ ∧ e3+ , e−φ = e−φ
√
b2 + r22r23
(3.40)
The T-dual vielbeins are7
e2± =±r23 e2−b
r2r3
e3
|h+B|f= r2
±r23 dx2 − b dx3
|h+B|f
e3± =±r22 e3+b
r3r2
e2
|h+B|f= r3
±r22 dx3 + b dx2
|h+B|f, (3.41)
ei± = ei , i 6= 2, 3 , (3.42)
5The vielbeins are computed by inverting (2.22), where we read a, b from the O(6, 6) matrix generating theT-duality action of this example
TV1TV4
=
„
a b
b a
«
, a =
„
13 −m 00 13 −m
«
b =
„
m 00 m
«
, m ≡
0
@
1 0 00 0 00 0 0
1
A .
6Strictly speaking the following is not an SU(3) structure, since B ∧Ω 6= 0. However, we can add for examplean e5 ∧ e6 component to make B proportional to Re(z2 ∧ z3). Since we will perform T-dualities in ∂2, ∂3, theadditional component would play no role, and stay unaffected by the T-duality.
7The dual vielbeins are again computed from (2.22) where the O(6, 6) matrix generating the T-duality actionfor this case is
TV2TV3
=
„
a b
b a
«
, a =
„
m 00 13
«
b =
„
13 −m 00 0
«
,
with m taking the same form as in footnote 5.
16
and |h+B|f = b2+r22r23. The T-dual structure is an SU(2) since, unlike the case in the previous
example, there are relative signs between J+ and J−:
J± = e1± ∧ e4± + e2± ∧ e5± + e3± ∧ e6±
= e1 ∧ e4 +1
|h+B|f
(
±r23 e2 ∧ e5 ± r22 e
3 ∧ e6 + b r3r2
e2 ∧ e6 − b r2r3
e3 ∧ e5)
,(3.43)
and similarly for Ω+, Ω−. Because of these relative signs, the SU (3)× SU (3) structure definedon E reduces to a local SU(2) on TM . For b = 0, the T-dual structure would be a “staticSU(2)” (k‖ = 0), in agreement with the examples studied in [28]. The effect of b in this case isto rotate this structure to a “dynamic SU(2)”, with k‖, k⊥ 6= 0.
Relation to non-geometry
All the discussion thus far has really been local: we have essentially used O(d, d) transformationson generalized spinors to map one local supergravity background into another. More generallyone is interested in whether these local geometries can really be completed into sensible globalstring backgrounds. It is well known that performing T-dualities on compact backgrounds withflux can lead to non-geometrical dual backgrounds. Non-geometry is an essentially stringyphenomenon so we cannot expect to see it directly in the supergravity description. In ourcontext this relates to the fact that T-duality does not act locally on the T 2 fibres. Nonethelesswe see that our examples do reflect elements of the non-geometry when one simply takes intoaccount that the base of the fibration is compact.
It is well known that a simple way of generating non-geometrical backgrounds is to take theT-dual of a T 2-fibration with non-trivial B-field on the fibre directions. This is precisely thecase we have considered in the previous examples.
By construction Φ± are independent of the fibre directions, as are Φ±. Thus effectively onemay ignore the fibre and simply consider the dependence of the pure spinors on the base. Ifb depends non-trivially on the base, in general, the original pure spinors are only defined onthe generalized tangent space E (2.1) twisted by the one-forms Λ(αβ) encoding the non-trivialpatching of the B field. Put another way, globally, in the expressions (3.30), the spinors Φ± aresections of S±(E) while Φ+
0 = e−iJ and Φ−0 = Ω are sections of the untwisted spinor bundles
S±(TM ⊕ T ∗M).Now consider the T-dual pure spinors. In general we will see that they are not well defined.
That is to say, they are not sections of S±(E) for some generalized tangent bundle E on thedual space. This is a reflection of the fact that the dual background is non-geometrical. To seeexplicitly that the spinors are not well defined, note that they can be written in terms of theoriginal ones as a β-transform (2.36)
Φ± = ∓i e−βΦ±0 , (3.44)
where the J and Ω defining Φ±0 take the standard form (3.30) but are evaluated using the basis
e1/r21, e2, e3, e4/r24 , e5, e6 for the non-type changing example, and e1, e2/r22 , e
3/r23 , e4, e5, e6
in the type-changing case. The bivector β is constructed from B by changing the form indicesinto vector indices, namely
β = b ∂/∂x1 ∧ ∂/∂x4 for non-type-changing ,
β = b ∂/∂x2 ∧ ∂/∂x3 for type-changing .(3.45)
Note that this is a completely generic feature of T 2 fibrations. Splitting the TM and T ∗Mbundles into base and fibre components one can write a generic B-transformation as the matrix
eB =
1 0 0 00 1 0 0
B(0) B(1)1 0
−B(1) B(2) 0 1
, (3.46)
17
where the B(0 is the component of B lying solely in the base, B(1) is the component with oneleg in the base and one in the fibre and B(2) lies solely in the fibre. If T represents the actionof T-duality on the T 2 fibre we have
eB 7→ TeBT−1 =
1 0 0 0
−B(1)1 0 B(2)
B(0) 0 1 B(1)
0 0 0 1
, T =
1 0 0 00 0 0 1
0 0 1 00 1 0 0
. (3.47)
Note that B(0) stays in the same position, i.e. in the T-dual setup is still a B-transform, whileB(1) and B(2) change positions. The former plays the role of a GL(d) transformation connectingthe base and the fiber, in agreement with (3.28), and the latter becomes a bivector just asin (3.45). We can now easily understand (3.44). In our T 2 examples we took B(0) = B(1) = 0.
Thus our orginal pure spinors could be written as eB(2)Φ±0 where Φ+
0 = e−iJ and Φ−0 = Ω. Then
T (e−B(2)Φ±0 ) = Te−B(2)
T−1TΦ±0 = e−βΦ±
0 . (3.48)
in agreement with (3.44).We now see the basic problem. If the original B-field on the fibres is non-trivial, the dual β-
transform will be similarly non-trivial. Put another way, if Φ± were sections of S±(E) where Eis patched over the base by B-transformations along the fibre directions, then Φ± are sectionsof some bundle were we must patch by β-transformations along the fibres. However, this isoutside the domain of conventional generalized geometry, where, by definition E can only betwisted by B-transforms. Hence Φ± appear to be not well defined.
Note that in the non-type changing case the problem is even more severe: not even the typeof the pure spinors is well-defined since eβ changes it. The problem is not simply that the typedepends on the location in the base, but rather that one cannot assign a unique type to thepure spinor at each point in the base. We also note that in both cases the metric defined bythe pure spinors is similarly ill-defined, as pointed out for example in [3] from Buscher rulesfor T-duality [23]. Again, the T-dual structure makes sense locally, but there is no good globaldescription.
One might have considered extending the notion of generalized tangent space to includeβ-transformations. The notion of such transforms was introduced in [18] and discussed in thephysics literature in [29] in the context of supergravity duals of deformations of conformalgauge theory, while their connection to non-geometry was explored in [8, 21]. At first sight,they seem as nice as B-transforms. In order to patch the T-dual bundle, one could try anduse the subgroup of O(d, d) built out of GL(d) and the β-transforms defined in (2.11). Thiswould correspond to identification of T ⊕ T ∗ with an extension of T ∗ by T via β-transform.However, unlike the B-transform extension this can prove problematic. Specifically there areno consistent gluing conditions on the two-fold overlaps that would satisfy cocycle conditions.This can be associated with an obstruction given by the first cohomology of the base H1(B,Z).We come back to this point in section 5.
3.4 Supersymmetric vacua and T-duality
T-duality is a powerful solution-generating tool for string theory backgrounds. Provided thestring background, that is the metric and the fluxes, has isometries, T-duality transformationsmap consistent string backgrounds into new consistent ones. At the level of supergravity, itmaps solutions of the supergravity equations of motion into new solutions. In this section wewill show that it also maps N = 1 supersymmetric backgrounds into N = 1 supersymmetricbackgrounds.
The necessary conditions for preserving N = 1 supersymmetry can be expressed as thethe closure of a pure spinor, and an integrability defect of its compatible partner which is
18
determined by RR fluxes. Clearly, if T–duality connects two supersymmetric backgrounds,they must separately satisfy the pure spinor equations.
The supersymmetry equations for N = 1 Minkowski vacua given in terms of the pure spinorswere found in [20] and read
d(e2AΦ1) = 0 , (3.49)
d(e2AΦ2) = e2AdA ∧ Φ2 +i
16
[
c−eAG+ ic+e
3A ∗E G]
. (3.50)
Let us explain the various ingredients in these equations. The pure spinors Φ1,2 are those of(2.44). The parity of Φ1 is the same as that of the RR fluxes, while Φ2 has the opposite parity.So Φ1(2) = Φ+(−) for type IIA, and the opposite for type IIB. c± are real constants that give
the relation between the norm of the 10D Killing spinors and the warp factor8, G = e−BF arethe RR field strengths with Bianchi identity dG = 0 in the absence of sources. The Hodgestar appearing in (3.49) is acting on sections of S±(E). It is related to the standard one actingon Λ•T ∗ as ∗E = e−B ∗ λ eB (where λ acts on p-forms by λG(p) = (−1)[p/2]G(p)) and it is thechirality operator on C+, as we will show later.
Note that the N = 1 supersymmetry equations are written precisely on the objets thattransform nicely under T-duality (the factor e2A does not play any role here, since as it is partof the space-time metric, it does not transform). We want to claim that these equations areinvariant under T-duality along a vector v that preserves the background, that is
LvΦ1,2 = 0, LvA = 0, LvG = 0 . (3.51)
Without loss of generality we can also take theB-field satisfying LvB = 0, so that the generalizedKilling vector is V0 = ∂/∂t+ dt, with v = ∂/∂t. We will show that if Φ1,2 are a solution to theequations (3.49), (3.50), their T-duals, Φ1,2, solve equations of the same form, with T-dual RRfield-strengths.
where we have added and subtracted i∂/∂tdΦ1,2 to build the Lie derivative along v of Φ1,2. Next,one can show that
− TV (dA ∧ Φ1,2) = dA ∧ Φ1,2 , −TV (G) = G , −TV (∗EG) = ∗EG . (3.53)
The first result is straightforward using (3.51), which implies ivdA = 0. The second one isprecisely the T-duality transformation of the RR fields. Indeed, the RR fields are O(d, d)spinors, and therefore transform as Φ, (3.12). The third equality needs a little more thinking.Inserting T 2
V = 1 we get
− TV ∗E TV TV G = (TV ∗E TV )G = ∗EG , (3.54)
where in the last equality we have used that ∗E = e−B ∗ λ eB transforms by conjugation. Thiscan be understood by noting that this combination is the chirality operator Γ+
(6) on C+. Indeed,
Γ+a in (2.37) acts on Spin(6, 6) spinors as
Γ+a ·G = ie+ a
G+ e+ a ∧G− ie+ aB ∧G
= eB(ie+ a+ e+ a ∧ ) e−BG ,
(3.55)
8More precisely, |η1|2 = c+eA + c−e
−A, |η2|2 = c+eA − c−e
−A. Backgrounds with D-branes and/or orientifoldplanes require |η1| = |η2| and therefore c− = 0.
which implies Γ+(6) = eB ∗λ e−B . Changing the sign of B (which is conventional, and could have
been taken opposite in (2.37)), this is just ∗E . Since the chirality operator transforms underO(d, d) by conjugation, we verify the last equality in (3.54).
We conclude that if Φ1,2 are pure spinors of an N = 1 vacuum, their T-duals Φ1,2 are purespinors of a vacuum with T-dual RR fields.
4 Generalized charges and the Courant bracket
One of the goals of this paper is to see how aspects of non-geometry might be encoded in thelanguage of generalized geometry. In the previous section we saw examples of non-geometryappearing as a result of T-duality on backgrounds with SU (3)× SU (3) structures. Specifically,assuming a torus fibration and restricting to the base of the manifold, we saw that the corre-sponding pure spinors Φ± were no longer sections of S±(E). Instead, to be globally definedon the base, one had to patch by elements of O(d, d), namely a β-transform, not contained inGgeom.
We now turn to a related problem. It has been argued that at the level of the effectivetheories non-geometric backgrounds are characterised by certain charges, or non-geometricalfluxes, Q and R. These are the analogues, and T-duals, of the “geometrical fluxes”, the H-fluxand the structure constants f of twisted torus compactifications. One way these fluxes appearis as structure constants in the 2n-dimensional Lie algebra of the effective gauged supergravitytheory [2, 3]. These can also be derived using world-sheet Hamiltonian methods [30,31].
Alternatively, they appear to define a sort of generalized derivative operator on sums offorms [5],
D = H ∧ + f · +Q · +Rx . (4.1)
Here H ∈ Λ3T ∗M , f ∈ TM ⊗Λ2T ∗M , Q ∈ Λ2TM ⊗T ∗M and R ∈ Λ3TM , and the action of fand Q on forms is by contraction on vector indices and antisymmetrization on the form indices.To date these charges have only been identified for very specific backgrounds.
In this section we propose a generalized geometrical definition of the generalized charges,as well as the operator D for generic backgrounds. That such a formulation exists is alreadysuggested by the fact that D can be interpreted as an operator on generalized spinors, since theseare sums of odd or even forms. We shall define the charges using the Courant bracket (2.13) andargue that they can be interpreted as components of a generalized spin connection [33]. A keypoint is that, as such, they will be gauge dependent, taking different values depending on theparticular generalized vielbein one uses. We make the connection to various specific examples,and discuss the global issues in the following section.
4.1 The Lie bracket and the spin connection
In conventional differential geometry the Lie bracket is dual to the exterior derivative in thesense that one can always be defined in terms of the other. In particular, given a form α one
This relation implies there are two equivalent ways of defining the spin-connection. Given anyframe ea and its inverse ea we can define the objects fa
bc in two different ways
[ea, eb] = f cabec , ⇔ dea = −1
2fabce
b ∧ ec . (4.3)
If ea are vielbeins for some metric, then the requirement that the Levi–Civita connection ismetric compatible and torsion-free implies that we can define the spin connection in terms offa
bc asωab =
12 (fcab + facb − fbca) e
c , (4.4)
where we have raised and lowered frame indices with frame metric δab.
4.2 Generalized charges, brackets and a generalized spin connection
The expression for ωab in terms of the Lie bracket, suggests that, in generalized geometry, one
can use the Courant bracket (2.13) to define a generalized spin connection [33]. Suppose wehave a basis given by the generalized vectors EA with A = 1, . . . , 2d, and we use the conventionswhere η and H take the form (2.23), or equivalently
η = 12 (Ea ⊗ Ea + Ea ⊗ Ea)
H = 12
(
δabEa ⊗ Eb + δabEa ⊗ Eb)
.(4.5)
Here we have split EA = (Ea, Ea) with a = 1, . . . , d. In the language of ref. [33] this has givenus a split of the generalized tangent space E = C0 + C⊥
0 spanned by Ea and Ea respectively. Itrequires that the resulting maps from C0 and C⊥
0 to TM and T ∗M are non-degenerate.In analogy to (4.3) one can then define
[EA, EB ] = FCABEC . (4.6)
Our claim is that the components of FABC are the generalized fluxes f , H, Q and R. To see how
this might work, let us first consider some special cases. If e+ = e−, the generalized vielbeinscan be written as (2.26) so that
Ea = ea , Ea = ea − ieaB . (4.7)
It is then easy to calculate[Ea, Eb] = f c
abEc −HabcEc ,
[Ea, Eb] = −f bacEc ,
[Ea, Eb] = 0 ,
(4.8)
where fabc is defined as in (4.3) and
Habc = −3(ie[cdBab] + fd[abBc]d) . (4.9)
One could also choose a basis based on the β-transform (2.11) where
Ea = ea + β · ea , Ea = ˆea , (4.10)
and, in order to reproduce the generalized metric (2.16), we have eaebδab = g
g = g −Bg−1B ,
β = −g−1Bg−1 .(4.11)
21
One then finds that[Ea, Eb] = f c
abEc ,[Ea, Eb] = −f b
acEc +QbcaEc ,
[Ea, Eb] = QabcEc +RabcEc ,
(4.12)
where fabc is defined as in (4.3) but using ea and
Qabc = iˆecdβ
ab + βadf bcd − βbdfa
cd , (4.13)
and
Rabc = βadiˆeddβbc − βbdiˆeddβ
ac + βadβbef cde , (4.14)
where β = 12β
ab ˆea ∧ ˆeb. Note that the new terms in the algebra only vanish if β = 0, showingthat in contrast to a closed B-transform, a constant β-transform is not an automorphism of theCourant bracket. For special cases though, the contractions of β on f appearing in (4.13) and(4.14) vanish, as we will show in section 5.1.
The general case is as follows. First note that Em(0) = dxm and E(0)m = ∂m is a (local) frame
for the O(d, d) metric η, but not in general for H. (Note that in this coordinate frame F = 0.)This implies that any given frame E can always be written as an O(d, d) rotation of E(0), thatis E = E(0)O or in components
Ea = Aamdxm + Bam∂m , Ea = Camdxm +Da
m∂m . (4.15)
The splitting condition implies that Aam and Da
m are non-degenerate. This then leads to thegeneral algebra
[Ea, Eb] = f cabEc −HabcEc ,
[Ea, Eb] = −f bacEc +Qbc
aEc ,[Ea, Eb] = Qab
cEc +RabcEc .(4.16)
where the fluxes f , H, Q etc are given in terms of derivatives of OAB .
The commutators (4.8) and (4.12) agree in form with those appearing in the effective gaugedsupergravity theories. However, there the elements EA are symmetry generators rather thangeneralized vectors. In addition, it is clear that our definition of FA
BC is gauge dependent.The frame EA is not uniquely defined; instead equivalent frames are related by O(d) × O(d)transformations (2.25). Changing frame thus changes the charges f , H, Q and R. A veryexplicit example is provided by the two frames (4.7) and (4.10). Both define the same generalizedmetric, but lead to very different charges (4.8) and (4.12). In fact there is a stronger statement.Locally, one can always make an O(d) × O(d) transformation to go the basis (4.7). Thus itwould appear that locally the Q and R charges can always be gauged away. As such it wouldseem hard, locally, to decide when a given set of charges implies we have a non-geometricalbackground and when not. We return to these points in the next section.
We want now to make the connection between the FABC and the generalized derivative D
given in (4.1). Hitchin [32] has noted that, in analogy to the duality between the Lie bracket andthe exterior derivative, the Courant bracket is dual to the action (2.30) of exterior derivativeon S±(E). Explicitly, if X · Φ is the Clifford action of X ∈ E on an spinor Φ, then
[X,Y ] · Φ = 12d [(X · Y − Y ·X) · Φ]+X · d (Y · Φ)− Y · d (X · Φ) + 1
2 (X · Y − Y ·X) · dΦ .(4.17)
This suggests that the charges FABC can be equally well defined using O(d, d) spinors. To
see how this works we need to consider what we mean by a generalized connection. Given
22
a tensor bundle W , the ordinary Levi–Civita connection ∇ = ∂ + ω is differential operator∇ : C∞(W ) → C∞(TM∗ ⊗W ). By analogy [33] a generalized connection is an operator9
D : C∞(W ) → C∞(E ⊗W ) , (4.18)
where W is some vector bundle which carries a representation of O(d, d). Again we can thinkof D as D = ∂ + , where the ordinary derivative ∂ simply gives a term in the T ∗M part of Eand nothing in the TM part. Thus one defines the derivative D, acting on a generalized vectorX = XAEA, as
DX =(
dXA +
ABX
B)
⊗ EA . (4.19)
Given a generalized connection one can then ask if it is compatible with η or with the generalizedmetric H, that is Dη = 0 or DH = 0, and if it is torsion free in a generalized sense. Forinstance, in [32, 33] a natural η and H compatible connection is defined, which is not torsionfree. If in particular one has a generalized connection that preserves the metric η, one can definea derivative of Spin(d, d) spinors by using the gamma matrices ΓA associated to a particularframe EA, that is
DAΦ = ∂AΦ+ 14A
BCΓBCΦ . (4.20)
If we now return to the exterior derivative we recall that it acts as a Dirac operator on theSpin(d, d) spinors, d : S±(E) → S∓(E). In the particular basis (2.27), where the generalizedvielbein takes the form (Em
(0), E(0)m) = (dxm, ∂m) we can write the exterior derivative in termsof a generalized η-compatible connection D
/DΦ = ΓADAΦ = Γm∂mΦ = dΦ . (4.21)
In this basis the spin-connection vanishes, consistent with the fact that FABC = 0. As we
commented above, a general frame, which is also a basis for H, can be written as an O(d, d)rotation of E(0). In this basis is non-zero and the Dirac operator can be written as
/DΦ = Od(
O−1Φ)
= dΦ+(
OdO−1)
Φ (4.22)
where by construction, we are writing the spinor Φ in a frame associated to EA, that is, whereit can be written as a tensor product of two spinors as in (2.40). Thus, for instance, if EA takesthe form (2.26), then we write elements of Φ in terms of the ea basis10
We see the appearance of the generalized fluxes f and H in the definition of /D just as inthe definition of D given in (4.1). In a more general basis one would also generate Q and Rterms. This is reflecting the duality between the exterior derivative and the Courant bracket.In summary, we see that the derivative D is simply the exterior derivative written in a frameadapted to the generalized vielbein EA.
9Note that we can use the metric η to identify E and E∗
10For simplicity we ignore the subtleties associated to the dilaton here.
23
5 Global properties, generalized charges and non-geometricity
In the previous section we proposed a generalized geometric expression for the charges f , H,Q and R, which arises from the Courant bracket between generalized vielbeins EA. This was apurely local notion. Crucially, it is also gauge dependent: changing the frame EA changes thecharges. A clear example was provided by the two bases (4.7) and (4.10). In fact, locally onecan always choose the gauge where EA takes the form (4.7) for which only the geometrical fand H charges appear. This implies that if Q and R are going to encode non-geometry, we canonly see this globally : there must be some global obstructions to gauging them away.
In this section we try to address this issue in some particular cases. We will focus onbackgrounds which admit a Td action. We give first the general analysis and then focus on twoknown examples which can lead to non-geometry. The advantage of such backgrounds is thatthe local fibration structure picks out a preferred frame EA with respect to which one can definethe charges, which allows us to see how non-geometry can be characterized in terms of Courantbrackets. Of course, for non-geometrical backgrounds, the Td fibration will not patch to forma proper manifold. As such it cannot be described using supergravity. Nonetheless we will seethat the twisting of the frame over the base of the fibration can be used to characterize the factthat the background is non-geometrical.
The existence of a preferred frame generically implies an additional structure beyond O(d, d).The extreme case of this are “generalized parallelizable” backgrounds, where, in analogy withconventional parallelizable manifolds, there is a globally preferred frame EA. We end the sectionwith a brief generic discussion of such backgrounds with additional structure.
5.1 Generalized charges and fibrations
In Section 4.2 we showed how the two different choices of bases for the generalized vielbeins, (4.7)and (4.10), give rise to two different algebrae with charges f and H, and Q and R, respectively.Here we consider a particular realisation of these two bases. More precisely we consider a classof manifolds which are Td fibrations. The structure of the metric and B-field is the same as inSection 3.3.1, but now the dimension of the fibre is d rather than three.
If the metric admits a Td action, the generalized vielbeins can be written as
(
Ea
Ea
)
=
ea′
i 0 0 0λa
i eaα 0 0
Ba′i Ba′α ea′i λa′
α
Bai Baα 0 eaα
dxi
dyα
∂i∂α
. (5.1)
In order not to clutter the expression above we defined the connections λai = eaαλ
αi and
λa′α = −ea′
iλiα. Similarly the components of the B-field are
Ba′α = ea′iBiα Ba′i = ea′
j(−Bij +Bjαλαi − λj
αBαi) , (5.2)
Baα = eaβBβα Bai = −ea
α(Bαβλβi +Bαi) . (5.3)
As we see, the vectors on the base are shifted by the derivatives along the torus due to thenontrivial fibration. It is straightforward to check that the generalized vielbeins (5.1) satisfythe algebra (4.8), where, because of the isometries in the fibre directions, the only non trivialcomponents of f are fa′
b′c′ = ieb′ iec′dea′ and fa
b′c′ = −e[b′iec′]
j∂iλaj .
As we will see in the examples below, there are at least two different ways to obtain the β-transformed basis by O(d, d) transformations. One possibility is to consider the torus fibrationwith a B-field with components in the fibre direction only, and to apply T-duality along the fibre.Since the torus directions are isometries, this is a perfectly lecit transformation. Alternativelywe can set the B-field to zero and perform a β-deformation on the metric respecting the torus
24
action. In both cases the resulting vielbein has the form
(
Ea
Ea
)
=
ea′
i 0 0 0λai 1 0 βaα
0 0 ea′i λa′
α
0 0 0 1
dxi
dyα
∂i∂α
, (5.4)
with βaα = eaβββα. This generalized vielbein gives the algebra (4.12). Note that the derivatives
along the fiber coordinates as well as the contractions of β and fab′c′ vanish. Moreover the
algebra (4.12) takes the canonical form, with Rabc = 0 and the only non-vanishing componentof Q-charge being Qab
c′ = iˆec′dβab = ∂c′β
ab.
Note that a corollary of the above computation is that, on a manifold that admits a T
d
action, a constant β-transform with components only along the Td fibre is a symmetry of theCourant bracket.
On the other side, it is not hard to see that when β lies along the fibers, the R-charge isnon-vanishing only if β depends on the torus coordinates. In our context, such a situation canarise when the B-field does not respect the isometries of the background.
As we already discussed, it is always possible to perform a local O(d)×O(d) transformation,(2.25), which preserves the form of the generalized metricH, (2.16), and maps the β-transformedbasis into the usual (B-transformed) basis on E
K E =1
2
(
O+ +O− O+ −O−
O+ −O− O+ +O−
)
eBeF eFβ
eTBeTF
=
eBeF
eTB−ˆeTFB
ˆeTF .
(5.5)
where the explicit expression for the matrices O± is
O+ = I O− =
(
1(eTF + eFβ)(e
TF − eFβ)
−1
)
. (5.6)
Let us examine the global issues associated with such a transformation. As explained earlier(see (2.14)) theB-field is defined only locally. Moreover Bαβ must not be a single-valued functionin order for the H-flux to be non-trivial in cohomology. This in turn means that the matrixO and the resulting generalized vielbeins are not single-valued either. As a consequence thetransformation in question, while not changing the generalized metric locally, cannot producea well-defined metric. Put differently, Qab
c′ and Hc′ab can be deformed into each other byusing local diffeomorphisms, provided they are exact. The difference in the vertical components(the position of the a, b indices) is not important here - the obstruction is given by the firstcohomology of the base of the torus fibration: when the first cohomology of the base is trivial,there simply do not exist any Bab which are not single-valued. This agrees with the T-dualityobstruction derived form the world-sheet perspective [14,15].
This is a general feature of the algebrae obtained from the generalized vielbeins: the Qcharges can be gauged away and the algebra can be smoothly deformed into a conventional onewith H and f (4.8), only if the first cohomology of the base is trivial.
5.2 Examples
In this section we illustrate with two basic and well known examples the general discussion above.The first one is probably the simplest and best known example of non-geometric background,namely the T-dual of the three-torus with a B-field along the T-duality directions. In this casethe base of the fibration is not simply connected and we will see that the local transformationthat should gauge the Q-charges away does not make sense globally.
25
The second example is the Lunin-Maldacena solution [34]. This is a good geometric back-ground obtained via β-transformation along the directions of the T 2 fiber. In this case we willsee that the Q-charges can indeed be gauged away by a good O(d)×O(d) transformation.
Three torus with H-flux
In this subsection we shall illustrate the construction on the prototypical example of a non-geometric background: the T-dual of the straight three-torus T 3 (Vol(T 3) = dx1 ∧ dx2 ∧ dx3)with a non-trivial NS three-form flux, H = kdx1 ∧ dx2 ∧ dx3. As we will see, this is an exampleof a general parallelizable manifold.
Clearly there is a basis of well-defined vectors ∂1, ∂2, ∂3 and a basis of one-forms dx1, dx2, dx3
on the tangent and the cotangent bundle, respectively. We choose a gauge where B = kx1dx2∧dx3. It is not hard to see that a global basis for the sections on E is given by
Note that this is of the standard triangular form (2.26). Calculating the Courant bracket yieldsthe familiar algebra
[Ea, Eb] = −HabcEc ,
[Ea, Eb] = 0 , where H123 = k ,
[Ea, Eb] = 0 .
(5.8)
One can now act on the basis by an element of O(3, 3) to go to the T-dual configuration. Tduality in the direction x3 amounts to ∂3 ↔ dx3. In order for the new basis to be split (that isfor the projections from C0 and C⊥
0 to TM and T ∗M to be non-degenerate) we have to performa local O(d)×O(d) transformation of the same form as the T-duality one. In this case this endsup in a relabeling of the vielbeins. We then arrive at the basis
where we have suppressed the tildes on the dual coordinates in order not to clutter the notation.Again this takes the standard form (2.26). It is well known that the dual background is a twistedtorus with zero B-field. This is reflected in the fact that the new basis consists of well-definedsections of T and T ∗. Computing the Courant bracket gives simply
[Ea, Eb] = f cabEc ,
[Ea, Eb] = −f bacEc , where f3
12 = k ,
[Ea, Eb] = 0 .
(5.10)
where we recognize the nilpotent Heisenberg algebra given by the structure constants (0, 0, 12).
The second T–duality – now in direction x2 – acts very much the same way and amounts to∂2 ↔ dx2. The new basis (again after some relabeling) is
( ˜Ea, ˜Ea) = (∂1, ∂2, ∂3; dx1, dx2 + kx1∂3, dx
3 − kx1∂2) (5.11)
and yields an algebra
[ ˜Ea, ˜Eb] = 0 ,
[ ˜Ea, ˜Eb] = −Qbca˜Ec , where Q23
1 = k ,
[ ˜Ea, ˜Eb] = Qabc˜Ec .
(5.12)
26
We now note that the new basis is not in the standard form (2.26). In fact, it is not a section ofE for any choice of extension T ∗M → E → TM . Rather, it is an extension of TM over T ∗M .
This is reflected in the fact that the generalized metric H built from ˜EA is not single valued asa function of x1. We are used to this happening because B is not single valued, but here themonodromy in H is a β-transformation rather than a B-transformation.
One can, of course, find a local map O(d) × O(d) map to put the basis (5.11) into thestandard form (2.26). Explicitly, in (2.25) one takes
However O− and hence K are clearly not single-valued, since x1 is periodic. Thus althoughlocally we can gauge the Q in (5.12) away (and replace it with f and H) we cannot do thisglobally.
This background is the simplest example of non-geometrical compactification, where the T 2
fibres, labeled by x2 and x3, are patched by a T-duality as one moves around the base S1,labeled by x1. As such, it is not a manifold since T-duality does not map points to points on
the fibres. It is therefore hard to define in what sense the basis ˜EA is global. Nonetheless, thebase S1 is still a conventional manifold, and we can simply imagine restricting everything to
this S1 (or equivalently, ignoring the fact that the fibres are compact). The ˜EA are then a globalbasis for the restricted generalized tangent space over S1. We also have the restrictions of TMand T ∗M . The statement that Q cannot be gauged away then has a well defined meaning interms of the restrictions, even if we cannot define the full compactification as a manifold.
We have seen that each of the three backgrounds, related by T-duality, are parallelizable inthe sense that one can introduce a globally defined basis EA. The three algebras (5.8), (5.10)and (5.12) are actually equivalent: the only difference is the split of the basis into Ea and Ea,which is related to how TM and T ∗M embed in E. The non-geometry of (5.12) was encoded inthe fact that the Q-charge could not be gauged away. Equivalently E, or rather its restrictionto the base S1, could not be viewed as an extension of TM by T ∗M . Put another way, itsstructure group was not in the Ggeom subgroup of O(3, 3).
The Lunin-Maldacena solution
The Lunin-Maldacena solution corresponds to a deformation of AdS5 × S5 that was originallyobtained by applying a T-duality, a rotation and a further T-duality on a T 2 inside S5 [34].AdS5×S5 can be written as a warped product of 4-dimensional Minkowski and the 6-dimensionalflat metric. Defining the three complex coordinated on R
6 as zi = µieiφ 11, the 6-dimensional
metric can be written as a (trivial) T 3 fibration
ds2 =
3∑
i=1
(dµi)2 + µ2
i (dφi)2 . (5.16)
11The coordinates µi are defined in terms of angles:
µ1 = cosα, µ2 = sinα cos θ, µ3 = sinα sin θ, (5.15)
27
As shown in [29,35], in this notation, the chain of transformations leading to the LM back-ground is equivalent to a β-deformation. In particular one can act on the generalized vielbeinwith the β-transform (2.11), where
β = γ
0 1 −1−1 0 11 −1 0
, (5.17)
where γ is the deformation parameter. Explicitely
E ′ = OE =
1
1 β1
1
eBeF
eBeF
=
eBeF eFβ
eBeF
, (5.18)
where eB = ea′
= δa′
idµi and eF = ea = δaiµidφi are the vielbeins of the flat metric (5.16) and
eFβ = γ
0 µ1 −µ1
−µ2 0 µ2
µ3 −µ3 0
. (5.19)
From the generalized metric H it is easy to see that the new metric and B-field are indeedthose of the LM solution
ds2 =3∑
i=1
(dµi +Gdφi)2 + γ2G(µ1µ2µ3)2(
3∑
i=1
dφi)2 , (5.20)
B = γG[(µ1µ2)2dφ1 ∧ dφ2 + (µ2µ3)
2dφ2 ∧ dφ3 + (µ3µ1)2dφ3 ∧ dφ1] , (5.21)
with G = [1 + γ2((µ1µ2)2 + (µ1µ3)
2 + (µ2µ3)2)]−1.
As in the previous example we can find an O(d)×O(d) transformation bringing the gener-alized vielbein to the triangular form (2.26). In this case (5.6) takes the form
O+ = I O− =
(
1GOF
−
)
(5.22)
with
OF− =
1− γ2(µ21µ
22 − µ2
2µ23 + µ2
1µ23) 2γµ1µ2(1 + γµ3) 2γµ1µ3(−1 + γµ2)
2γµ1µ2(−1 + γµ3) 1− γ2(µ21µ
22 + µ2
2µ23 − µ2
1µ23) 2γµ2µ3(1 + γµ1)
2γµ1µ3(1 + γµ2) 2γµ2µ3(−1 + γµ1) 1− γ2(µ22µ
23 + µ2
1µ23 − µ2
1µ22)
(5.23)Differently from the previous example, the transformation O− does not contain any non-
single valued function of the base. This is also related to the fact that since we have a simplyconnected base it is not possible to choose a B-field with two legs along the fibre to be notsingle valued.
5.3 Generalized parallelizable backgrounds
The simplest way around the gauge-dependence of the charges F is to assume that there issome preferred frame EA, and to define F as the values in this frame. In the previous examples,such a class of frames was defined by the Td fibration structure. In particular, for those basedon the three-torus with H-flux, there was actually a fixed globally defined frame. This is anexample of a “generalized parallelizable” background. In this section, we would like briefly toaddress some of the constraints on the generic form of the local geometry of such backgrounds,
28
and in particular ask what charges F can appear. We will also see how T-duality acts on suchbackgrounds.
Recall that in conventional geometry on a parallelizable manifold there exists a globallydefined frame ea implying the tangent bundle TM is trivial. In addition one can further assumethat the manifold admits a metric of the form g = gabe
a ⊗ eb with gab constant. (In themathematics literature this is known as “consistent absolute parallelism” [36, 37].) Except forthe special case of S7, the manifold is then a Lie group and the functions fa
bc are the structureconstants. In complete analogy one can define a “generalized parallelizable compactification”where there is now a globally defined frame EA of E. We will also assume that the O(d, d)metric takes the form (2.23), which we can also write as
η = ηABEA ⊗ EB , (5.24)
but drop the requirement that H takes a particular form. Thus the EA are defined up to globalO(d, d) transformations. Up to such rotations, there is then a unique set of charges defined by
[EA, EB ] = FCABEC , (5.25)
which are taken to be constant. Again, the notion of “globally defined” becomes unclear when wetalk about non-geometrical backgrounds. As it stands we will only assume such a local geometryand corresponding charges. The question of how these might complete into geometrical or non-geometrical backgrounds is not discussed. Note that such backgrounds are somewhat analogousto the general twisted double torus backgrounds discussed for instance in [10,38]. The differenceis that there the algebra is realized in terms of the Lie bracket of vector fields on a “doubled”2d-dimensional space. Here we are considering a more restricted example: we use the Courantbracket on generalized vectors on what is locally a conventional d-dimensional space.
The three-torus examples above are each generalized parallelizable manifolds. The algebrasof the EA are actually isomorphic in each case. It is the split of E into TM and T ∗M (andhence of EA into (Ea, Ea)) that gave the different interpretations of the structure constants ascorresponding to H, f or Q charge.
Let us see what conditions the existence of the algebra (5.25) realized by the Courant bracketplaces on the local geometry of the background. The first conditions follow from the fact thatwe can define the O(d, d) metric H as in (4.5). From Proposition 3.16 of [18] we see that, sincethe FA
BC are constant, the Courant bracket (5.25) on EA satisfies the Jacobi identity and hencedefines a Lie algebra h. Given Proposition 3.18 of [18], we also have
ηCDFDAB + ηBDF
DAC = 0 . (5.26)
This implies that the adjoint representation of the algebra (4.16), where the generators are givenby (TA)B
C = FCAB , acts as a sub-algebra h ⊂ o(d, d).
Next recall that under the projection π : E → TM the Courant bracket reduces to the Liebracket
π([X,Y ]) = [π(X), π(Y )]Lie . (5.27)
Writing vA = π(EA) this simply states that [vA, vB ]Lie = fCABvC . Thus there is a realization
of the algebra h in terms of 2d vector fields on M , though of course this may be somewhatdegenerate since some vA may vanish identically. Since the EA are a basis for E, the vA mustform a basis for TM , that is, there must be at least d non-vanishing vA at each point p ofM . Exponentiating the Lie algebra action into diffeomorphisms we see that M is locally ahomogeneous space, with an action of a group H with Lie algebra h. Let us fix some pointp ∈ M . If we identify X = XAEA with constant XA as elements of the Lie algebra h we definethe set of vectors X with vanishing π(X) at a given point p ∈ M
kp = X ∈ h : π(X)|p = 0 . (5.28)
29
This must be a d-dimensional subset of h. Since the Lie bracket of two vector fields that vanishat p ∈ M must itself vanish at p ∈ M , we see that kp must form a closed subalgebra. Hence wesee that locally M must be a coset space. We can write
M = H/K , K ⊂ H ⊂ O(d, d) , (5.29)
where H is a 2d-dimensional group with Lie algebra h given by (5.25) and K is a d-dimensionalsubgroup with Lie algebra isomorphic to kp. For a parallelizable manifold, M is (almost al-ways) locally a group manifold. Thus we see, as one might expect, generalized parallelizablecompactification appear to be more general.
Let us now turn to the fluxes F . At the point p, generalized vectors X ∈ kp lie solely inT ∗pM . Hence we can locally identify Ea as a basis for kp and, using the metric η, decompose
h = kp ⊕mp, with Ea a basis for mp. Hence, for any generalized parallelizable compactification,since kp is a closed subalgebra, we see that one cannot arrange all fluxes to be non-zero. Inparticular, one can always use a global O(d, d) rotation to align the basis EA such that Ea spankp and EA span mp and
Rabc = 0 . (5.30)
This is in agreement with the T 3 with flux examples discussed above.
In this discussion we have only considered some of the conditions on the generalized paral-lelizable background that follow from the Courant bracket structure. One would expect addi-tional conditions, such as compatibility with a generalized metric of the form H = HABEA⊗EB,and probably a curvature condition as in [10]. It would also be interesting to find specific ex-amples where M is indeed locally a coset rather than a group manifold as in the T 3 with H-fluxexamples.
Let us end this section by discussing the generic action of T-duality on generalized par-allelizable backgrounds. Suppose we have a generalized Killing vector V which preserves theparallelizable structure, that is
LV EA = 0 ∀A . (5.31)
We can always normalize V such that η(V, V ) = 1 and define the T-duality operator TV asin (3.9). Using the general relations [18]
LXY = [X,Y ] + dη(X,Y ) ,
iπ(X)dη(Y,Z) = η(LXY,Z) + η(Y,LXZ) ,
[X, fY ] = f [X,Y ] + (iπ(X)df)Y − η(X,Y )df ,
(5.32)
and the fact that η(EA, EB) = ηAB and η(V, V ) = 1 are constant, it is relatively straightforwardto show that
[TV EA, TV EB ] = [EA, EB ]− 2(
iπ(EA)dη(V, EB)− iπ(EB)dη(V, EA))
V
= TV [EA, EB ] .(5.33)
Thus we see that TV is an automorphism of the generalized parallelizable algebra. A particularexample is the fact that the three algebras arising from T-duality of the T 3 with H-flux are allisomorphic. They are of course distinguished by the way one identifies vectors and forms in E.
Acknowledgments
We thank Dima Belov, Gianguido D’Allagata, Nick Halmagyi and Andrei Micu for useful dis-cussions. This work is supported in part by RTN contracts MRTN-CT-2004-005104 and MRTN-CT-2004-512194 and by ANR grants BLAN06-3-137168 (MG and RM) and BLAN05-0079-01(MP).
30
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